Von Neumann Type Inequalities for NC Operator Balls

We establish finite-dimensional von Neumann type inequalities for commuting matrix tuples in noncommutative operator balls. Let $\mathrm{\Omega }={\mathbb{D}}_Q$ be the NC operator ball of a linear pencil $Q$ with linearly independent coefficients. If $\mathrm{\Omega }(1)$ admits a transitive subgroup of automorphisms with NC lifts and $\mathrm{\Omega }(1)$ has the bounded $H^{\mathrm{\infty }}$-Gleason property at the origin, then $\mathrm{\parallel }f(T)\mathrm{\parallel }\le C_{\mathrm{\Omega },n}\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{\mathrm{\infty }}(\mathrm{\Omega }(1))}$ for every commuting tuple $T\in \mathrm{\Omega }(n)$, with $C_{\mathrm{\Omega },n}=(2{\rho }_{\mathrm{\Omega }}{K}_{\mathrm{\Omega }}{)}^{n-1}$; here $K_{\mathrm{\Omega }}$ is the Gleason constant and ${\rho }_{\mathrm{\Omega }}$ is the sum of the suprema of the coordinate functions on $\mathrm{\Omega }(1)$. The scalar coordinate bounds persist at every matrix level. The hypotheses hold for products of row and column balls, where ${\rho }_{\mathrm{\Omega }}=d$, yielding an unconditional estimate $(2d{K}_{\mathrm{\Omega }}{)}^{n-1}$. For a general type I Cartan domain $R_{p,q}$, we obtain only a conditional result: the same bound holds provided the scalar domain has the bounded $H^{\mathrm{\infty }}$-Gleason property at the origin, which we do not establish when $\min (p,q)\ge 2$.